Determining XY coordinates from quadrilateral measurements

I'm trying to determine the XY coordinates of 3 corner points of a quadrilateral shape based on known lengths of the sides of that shape. I know the lengths of all 4 sides of the shape as well as the lengths of both cross lengths (effectively making two adjacent triangles). I've attached a picture to indicate the lengths of the triangles / quadrilateral shape as well as the corner of the shape that would serve as the origin of the system. I would like to find the coordinates of the 3 other corners of the shape in respect to the known XY coordinates of the origin. Any assistance would be greatly appreciated.

Surely if you got a bunch of rods with the lengths given, you would rebuild the shape in only one way?

It is over-determined but it can be solved directly from a subset of the information.
The main problem is that the axis choice is underspecified. You also need to know the orientation of the axis wrt the figure.

Where is the need for a statistical method?
(Trilateration is for measuring with error - the uncertainties here are between 0.3% and 0.1% over 2-4m so maybe the 5mm uncertainty is important?)

It is a valid positioning system or method of solving a triangle, from whence its name.
Knowledge of the lengths of all three sides completely determine any triangle.
However, very often, redundant observations are taken to improve acuracy.

The measurements are given to the nearest cm - uncertainties are not specified (we don't know how accurate the measurements are[*]) so guess about 0.5cm[**] uncertainties.
That does seem a tad large ... but still of the order of 1:100. It may be easier just to remeasure to the nearest mm if more accuracy is needed.

We'll have to get feedback to see how important the uncertainties are to this particular problem.

It remains that it is possible to find the coordinates of the points directly from the measurements. If uncertainties are somewhat important, then they can be found from the undergrad rules of thumb.

I wonder how much extra accuracy you'd gain - there are three measurements associated with each point.

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[*] maybe the measurements were taken with a laser rangefinder between built-in reference studs: it just happened to come to whole cms to nanometer accuracy and OP did not feel the need to put in all the trailing zeros. Unlikely admittedly...
[**] over-estimate: it suggests that we think that 32% of repeat measurements would differ from this one by more than 5mm or something.

You can use trilateration for measurement without regard to error and still solve the triangles when you have measured all three sides.

Further as you point out we don't know the conditions of measurement of these distances since they could be slant measurements. The OP certainly doesn't mention that they are all in the plane of the XY axes.

You can use trilateration for measurement without regard to error and still solve the triangles when you have measured all three sides.

OK - the paper you reference stresses the error thing. If you do it without error, doesn't it reduce to triangulation?

Further as you point out we don't know the conditions of measurement of these distances since they could be slant measurements. The OP certainly doesn't mention that they are all in the plane of the XY axes.

Considering the context, not too unreasonable an assumption (all measurements in the same plane). Real life is messy. Shall wee see what OP says?

You know - if this is, as I suspect, something already built - you can measure the coordinates directly using a string grid. You can make a right-angle from a 3-4-5 triangle (tying string together is the usual way).